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This paper presents a novel method for determining the worm wheel tooth flank surface. The worm wheel model is developed according to the theory of gearing and the gear generation mechanism. Two cases are considered simultaneously. In one case, the worm wheel is generated by the hourglass worm hob with a rectilinear tooth axial profile, while in the other case, it is generated with an arc (concave and convex) tooth axial profile. The contact lines between the hourglass worm hob and worm wheel surfaces are determined. Surface formation generated during processing by the tool external edge and formation so-called region II was presented. The manner of separating regions on the worm wheel tooth flank surface was developed. Because the difference between the parameters that describe the tool side surface, only the flowchart of the algorithm for determining the worm wheel tooth side surface of rectilinear profile is presented. For the arc profile, the algorithm flowchart is analogous. The proposed procedures and the developed computer-aided worm wheel generating program allow obtaining complete mathematical representation of worm wheel tooth flank surface.

The paper presents the method of determining geometric contact pattern by using the direct computer-aided design (CAD) method for ideal globoid worm gear in which mounting deviations are considered. The tooth contact analysis was performed for all cycle of worm rotation. Based on the results of temporary contact pattern, graphical characteristics of the contact area size depending on worm position were made. A complete analysis of the correctness of gear meshing can be obtained based on presented method. If the worm or worm wheel is incorrectly designed in terms of the geometry, the meshing simulation of CAD models can indicate the collision. Geometric contact pattern analyses were made at two pressure angles of ideal gear. The analysis of the influence of mounting deviations was done against one selected pressure angle and one gear position.

In the article research related to the increasing in the scope of application of additive technologies in the construction of machines for the production of gears with involute and sinusoidal profile from polymeric materials was presented. The designed original research stand and with its use carried out a series of preliminary fatigue tests of the obtained polymer gears with an involute and sinusoidal profile was show. The tested gears were obtained in the technology of rapid prototyping by the FFF (Fused Filament Fabrication) method from polymer composites obtained in the form of an ABS (acrylonitrile/butadiene/styrene) – based filament. Based on the results obtained, it was noticed that the temperature in the meshing zone of the gears with involute profiles is higher than for sinusoidal profiles, regardless of the type of composite from which the gears were made. It should also be noted that in the range of nominal load, the sound intensity level of the meshing zone is also lower for gears with a sinusoidal profile than for gears with an involute profile. Based on the tests carried out, an increase in sound intensity was noticed in the case of gears obtained from selected composites (Table 2) compared to the gear obtained from unfilled ABS. However, in the case of gears obtained from the tested composite materials, we observe a decrease in the gear operating temperature (Table 2). The most favorable results of these tests were obtained for gears with a sinusoidal profile obtained from ABS + P2 composite, which show the lowest operating temperature of the gear. It should also be mentioned that slightly worse results were obtained for involute gears.

The paper presents the method of approximating curves with a single segment of the B-Spline and Bézier curves. The method for determining a single curve segment using the optimization methods in the CATIA environment is shown. The algorithms of simulated annealing and design of experiment are used for optimization. For the same purpose, a new original procedure for determining the distance between the given curves using explicit parameters in the CATIA environment was also used. This approximation of the cyclic curves results in the curve oscillation as shown in the examples. The results show that the approximation method with Bézier curve using control points as “free” points can be applied to obtain the best results of approximation.

In this article, a mathematical description of tooth flank surface of the globoidal worm and worm wheel generated by the hourglass worm hob with straight tooth axial profile is presented. The kinematic system of globoidal worm gear is shown. The equation of globoid helix and tooth axial profile of worm is derived to determine worm tooth surface. Based on the equation of meshing the contact lines are obtained. The mathematical description of globoidal worm wheel tooth flank is performed on the basis of contact lines and generating the tooth side by the extreme cutting edge of worm hob. The presented mathematical model of tooth flank of TA worm and worm wheel can be used e.g. to analyse the contact pattern of the gear.

Presented is the method of determining the contact pattern in globoidal worm gear using CAD environment. Contact pattern analysis was performed for the gear with straight tooth profile in the central plane. The influence of the pressure angle on the shape and size of the contact pattern was investigated.

Presented is a mathematical description of tooth flank surface of the wormwheel in globoid worm gear. The kinematic system of tooth formation was performed. The mathematical description of tooth flank of globoid worm was used.

This paper presents a mathematical model of globoid worm with concave and convex tooth profile. The method of creating the parametric equation of the profile was shown. The parametric equation of teeth surfaces of globoid worm with nostraight tooth profile was obtained.